The acoustic wave equation in the expanding universe. Sachs-Wolfe theorem

The Sachs-Wolfe theorem (Sachs and Wolfe, 1967, pp. 76-77) contains two separate theses formulated for two different equations of state: the first ((i) p. 76) for the pressureless matter (p = 0), and the second ((ii) p. 77) for ultrarelativistic gas (p = ǫ/3). The first thesis was recently recalculated and rediscussed by Sachs et al., (2007). In this paper we will concentrate on the second, and for the purposees of this paper we will call it the acoustic theorem to clearly distinguish between them both.

Acoustic theorem refers to the spatially flat (K = 0), hot (p = ǫ/3) Friedman-Robertson-Walker universe, and to the scalar perturbation propagating in it. It satates, that by the appropriate choice of the perturbation variable one can express the propagation equation in the form of d’Alembert equation in Minkowski space-time. Scalar perturbations in the flat, early universe propagate in a similar manner to electromagnetic or gravitational waves (Sachs and Wolfe, 1967, p. 79, see also Grishchuk 2007).

On the other hand, the d’Alembert equation describing scalar field in the dust (p = 0) cosmological model, can be transformed into the d’Alembert equation in the static Robertson-Walker space-time regardless of the universe space curvature (Klainerman and Sarnak, 1981). We may, therefore, suppose that the flatness assumption in the Sachs-Wolfe theorem is excessive and that the theorem is true in the general case.

Indeed, as we show below, the scalar perturbations reveal similar features in both, the flat and curved universes, i.e. they propagate as waves in the static Robertson-Walker space-time. The proof of this fact is the subject of this paper. The Sachs-Wolfe acoustic theorem may apply to the microwave background radiation (Sachs et al., 2007).

The paper consists of a brief description of the problem, and the short code in Mathematica. Throughout this paper c = 1 and 8πG = 1.

We adopt Robertson-Walker metrics in spherical coordinates x σ = {η, χ, ϑ, ϕ}

with the scale factor a(η) appropriate for the equation of state p = ǫ/3

Consider the scalar perturbations in the synchronous system of reference. The metric tensor correction is then determined by two scalar functions E and C (Stewart, 1990)

△ stands for the Beltrami-Laplace operator on the η = const hypersurface. Eventually, the metrics takes the form g µν = g (RW)µν + δg µν (5) In the first order of the perturbation expansion the Einstein equations with the metric tensor (5) reduce to

where we put µ = C, λ = -△E (to fit to (Lifshitz andKhalatnikov, 1963, Landau andLifshitz, 2000)). c(η) stands for sound velocity. The energy density contrast δ = δǫ/ǫ reads

Let us define a new perturbation variable Ψ with help of the second order differential transformation of the density contrast δ

The function Ψ (x σ ) is the solution of the d’Alembert equation

with the Beltrami-Laplace operator △ = (3) g mn ∇ m ∇ n acting in the space

Theorem 1 (Sachs-Wolfe acoustic theorem for K = 0, ±1). Scalar perturbations in the hot (p = ǫ/3) Friedman-Robertson-Walker universe of arbitrary space curvature (K = 0, ±1), expressed in terms of the perturbation variable Ψ (9) obey the wave equation ( 10) in the static Robertson-Walker space-time g = diag (-1, (3) g).

Proof.

We perform this calculation in Mathematica. The code is enclosed below, and also can be downloaded from: (* http://drac.oa.uj.edu.pl/usr/woszcz/kody/acoucticRWcode.nb ) ( Coordinates *) Clear[K, a, c, ǫ, H, Lap, Ψ] X = {η, χ, ϑ, ϕ}; XX = {η , χ , ϑ , ϕ }; x = Sequence@@X; xx = Sequence@@XX;

(* Friedman-Robertson-Walker background for ǫ = 3p *)

//Simplify;

(* Higher order derivatives *) ,d1 ,d2 ,d3 ) [xx] = D λ (2,0,0,0) [x], {χ, d1}, {ϑ, d2}, {ϕ, d3} ; µ (2,d1 ,d2 ,d3 ) [xx] = D µ (2,0,0,0) [x], {χ, d1}, {ϑ, d2}, {ϕ, d3} ;

(* Laplacian in the maximally symmetric 3-dim curved space *)

(* The gauge-invariant variable Ψ *)

(* The wave equation *)

By obtaining equation ( 10) we have expressed the problem of cosmological density perturbations in terms of the semiclassical theory (the field theory in curved spacetime -see Birrell and Davies, 1982). Consequently, the key facts resulting from the semiclassical theory apply also to the perturbed expanding universe.

Scalar perturbations in the early universe form the acoustic field. Acoustic waves are dispersed by the space curvature. The dispersion relation for the equation (10) takes the form (Birrell and Davies, 1982, eq. (5.27))

and the group velocity reads

In flat space (K = 0) the group velocity is constant and equal to 1/ √ 3. This fact has been already established by Sachs and Wolfe (Sachs and Wolfe, 1967).

Beyond the K = 0 case the dispersion relation is nonlinear. In the space of negative curvature K = -1 the waves behave as the scalar field with the mass m = 1. The group velocity is the function of k and decreases to zero in the k → 0 limit lim k→0 v g = 0 (14) while the limit frequency is still positive

Acoustic behaviour does not extend

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